and , while the phase diagram, for the isotropic Ashkin-Teller model,contains three phases, I.e., the fully disordered paramagnetic phase Para, the fully ordered ferromagnetic phase Baxter Phase, and the partially ordered ferromagnetic phase .
Using the recursion method, we study the phase transitions of the Ashkin–Teller model on the Bethe lattice, restricting ourselves to the case of ferromagnetic interactions. The isotropic Ashkin–Teller model and the anisotropic one are respectively investigated, and exact expressions for the free energy and the magnetization are obtained. It can be found that each of the three varieties of phase diagrams, for the anisotropic Ashkin–Teller model, consists of four phases, i.e., the fully disordered paramagnetic phase Para , the fully ordered ferromagnetic phase Ferro , and two partially ordered ferromagnetic phases ⟨σ⟩ and ⟨σ s ⟩, while the phase diagram, for the isotropic Ashkin–Teller model, contains three phases, i.e., the fully disordered paramagnetic phase Para , the fully ordered ferromagnetic phase Baxter Phase , and the partially ordered ferromagnetic phase ⟨σ s ⟩.
The phase transitions of the anisotropic Ashkin-Teller model on a family of diamond-type hierarchical lattices is studied by means of the transfer-matrix method and the real-space renormalization-group transformation. We find that the phase diagram, for the ferromagnetic case, consists of five phases, i.e., the fully disordered paramagnetic phase P, the fully ordered ferromagnetic phase F, and three partially ordered ferromagnetic phases F(s), F(sigma), and F(s sigma), as well as ten nontrivial fixed points. The correlation length critical exponents and the crossover exponents are also calculated. In addition, we also investigate the variations of the critical exponents with the fractal dimension d(f), the number of branches m, and the number of bonds per branch b of the generator of the family of diamond-type hierarchical lattices. Finally we give a brief discussion about universality.
By using Monte Carlo simulation,we have studied phase transitions of the isotropic Ashkin-Teller model,for the ferromagnetic case,on a two-dimensional rewiring-type small world network,and obtained phase diagrams for several different values of the rewiring probability.It can be found that the rewiring probability have little effect on the structure of the phase diagrams,each of which consists of three phases,i.e.,the fully disordered paramagnetic phase, the partially ordered ferromagnetic phase and the fully ordered ferromagnetic phase,and there are first-order transitions between the fully ordered ferromagnetic phase and the fully disordered paramagnetic phase as well as between the fully ordered ferromagnetic phase and the partially ordered ferromagnetic phase.
Using the real-space renormalization-group transformation, we study the phase transitions of the Ashkin-Teller model including the antiferromagnetic interactions on a type of diamond hierarchical lattices, of which the number of bonds per branch of the generator is odd. The isotropic Ashkin-Teller model and the anisotropic one are, respectively, investigated. We find that the phase diagram, for the isotropic Ashkin-Teller model, consists of five phases, two of which are associated with the partially antiferromagnetic ordering of the system, while the phase diagram, for the anisotropic Ashkin-Teller model, contains 11 phases, six of which are related to the partially antiferromagnetic ordering of the system. The correlation length critical exponents and the crossover exponents are also calculated.
In order to study the nature of phase transitions in magnetic systems, the aim of our research is to explore the S=1 and σ=3/2 states where the ferromagnetism as well as the influence of different crystal fields will be studied by the Monte Carlo method of The Ashken-Teller model which is a coupling of two Ising models with a four-component parameter. We have given some theoretical definitions related to the phase transition as well as the types of magnetism also the spin models and the simulation method. First, we identify the most stable phases in the phase diagrams at temperature T=0 by energy minimization. Then For higher temperatures, we find phase diagrams rich in ordered phases: a 3/2 Baxter phase and a 1/2 Baxter phase as well as a phase that appears neither in spin 1 nor in spin 3/2 ATM, and finally, a <σ> = 1/2 phases with 1st and 2nd order.